Generalized continua are increasingly used to account for size effects in the mechanics of materials and structures. They include higher grade media characterized by the presence of zeroth, first and second (or more) gradient of the strain tensor in the constitutive setting [1], and higher order continua that incorporate additional degrees of freedom like Cosserat rotations or micromorphic deformations [2]. In the recent years, stress gradient theories have emerged sometimes presented as special cases of strain gradient theories, or as fundamentally distinct theories [3,4]. This point of view will be developed in the presentation showing that strain and stress gradient models are NO dual theories but essentially distinct approaches to material behaviour.
A similar situation is encountered in the gradient of entropy and gradient of temperature theories of rigid conductors [5,6].
Stress gradient continua include a third order tensor of additional degrees of freedom, called microdisplacement, independently of the usual displacement field. This makes them akin to general micromorphic media introduced by Eringen and Germain [7]. The relations between such gradient and micromorphic continua will be depicted based on suited internal constraints [8].
Applications will be presented dealing with the regularization of stress singularities or concentration in structures and with the dispersion of elastic waves.

[7] P. Germain. The method of virtual power in continuum mechanics. Part 2 : Microstructure.

SIAM J Appl Math, vol. 25, pp. 556–575, 1973.

[8] S. Forest and K. Sab, Finite deformation second order micromorphic theory and its relations to strain and stress gradient models, Mathematics and Mechanics of Solids, in press, 2017. doi:10.1177/1081286517720844